function FVlinearadvectionQUAD1D 
% File name: FVlinearadvectionQUAD1D.m 
% Description:  Solves the single 1D PDE, dU/dt + div(H) = 0, 
% using the QUAD finite volume scheme on a uniform Cartesian mesh which is 
% a single row of cells in the x direction with sides dx and dy. 
% This equation corresponds to the finite volume form: 
% 
% doubleintegral(dU/dt dA) + lineintegral(H.n ds) = 0. 
% 
% In this case the equation is the linear advection equation so:  
% velocity: v = vx i + vy j where vx is constant and vy = 0. 
% flux density: H = vU = vx U i 
% 
% There are NI cells in the i (x) direction and 1 cell in the j (y) 
% direction. Since vy = 0 there is flow only through the left and right  
% sides of each cell. Cell areas and side vectors are the same for each cell. 
% 
% The program is written in a structured way using subfunctions so that it 
% can be modified easily to the pseudo-1D and 2D cases on non-Cartesian 
% structured meshes. 
% 
% Initial conditions:  Gaussian at cell centres. 
% 
% Boundary conditions: Dirichlet, u=0 in the 2 left and 1 right ghost cells.  
% 
% subfunctions: freadmesh, fcellarea, fsidevectors, finitialu, 
%               finterfaceflux, fghostcells, fcalcdt, fplotresults,  
%               fdrawmesh. 
% 
% Verified:  Not verified. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%% 
clc; clf; clear all; 
runtime=130.0;  % runtime in seconds 
t=0;          % current time 
timelevel=0;  % current time level 
vx=0.5;        % velocity component in x direction [m/s] 
vy=0;          % zero velocity component in y direction  
v=[vx,vy];     % velocity vector = vx i + 0 j 
[x,y,xcen,solid]=freadmesh; % gets coordinates of the 1D computational mesh  
                            % and solid flags (excluding ghost cells) 
% Note that there are no solid interior cells in 1D. 
disp('mesh read in') 
dy=y(1,2)-y(1,1); 
[m,n]=size(x);  
NI=m-1; % number of computational cells in i direction. 
A=fcellarea(x,y);   % computes constant cell area 
disp('calculated cell area') 
S=fsidevectors(x,y); % compute and store cell side vectors                                          
disp('calculated cell side vectors') 
u0=finitialu(xcen,t,vx); % Put initial cell centre values of u in a NIx1 array 
uinitial=u0; % store initial profile for plotting 
disp('inserted initial conditions') 
% Append extra cells around arrays to store any ghost values. 
u0=finsertghostcells(u0);  % u0 is now (NI+2)x1 
u1=zeros(size(u0));  % (NI+2)x1 array for u values at next time level 
%% 
disp('time marching starts')    
while(t<runtime)    
 timelevel=timelevel+1   
 u0=fbcs(u0);  % Implement boundary conditions using ghost cells.  
               % u0 is (NI+2)x1 and each computational cell is  
               % indexed i=2 to NI+1. 
 dt=fcalcdt(A,S,v); % Finds stable time step for each iteration. 
 t=t+dt  % update time   
   for i=3:NI+2  
       IH=finterfaceflux(v,u0,i); % gets the left and right interface fluxes for cell i 
       IHright=[IH(1,1),IH(1,2)]; % interface flux vector right side  
       IHleft=[IH(2,1),IH(2,2)];  % interface flux vector for left side 
       % 
       sright=[S(1,1),S(1,2)]; % side vector for right side 
       sleft=[S(2,1),S(2,2)];  % side vector for left side 
       % FV scheme 
       % compute total flux out of cell i 
       totalfluxout=dot(IHright,sright)+dot(IHleft,sleft); 
        
       % totalfluxout=vx*u0(i,1)*dy+vx*u0(i-1,1)*(-dy); 
       u1(i,1)=u0(i,1)-(dt/A)*totalfluxout;  
   end % of i loop 
 %        
   u0=u1;  % update u values 
end  % of while loop 
disp('1D Quad scheme ended at time t') 
t 
%fdrawmesh(x,y,solid) 
%fdisplay(u0); % print results for a small mesh as a matrix 
uexact=finitialu(xcen,t,vx); % exact solution 
fplotresults(xcen,u0,uinitial,uexact) % plot results 
disp('finished 1D QUAD scheme, see figures') 
end % FVlinearadvectionQUAD1D 
%%%%%%%%%%%%%%%%%%%%%%%%%%% subfunctions  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function [x,y,xcen,solid]=freadmesh 
% Verification:  not verified 
% The mesh is structured and has NI cells in the i (x) direction and  
% 1 cell in the j (y) direction. 
% The x and y coordinates of the lower left hand corner of cell (i,j) are 
% held in arrays x and y respectively which are both (NI+1)x2.  In 
% this way the 4 vertices of cell (i,j) are (x(i),y(j)), (x(i+1),y(j)), 
% (x(i+1),y(j+1)) and (x(i),y(j+1)).  solid is an NI by 1 array which  
% flags solid cells.  If cell (i,j) is solid then solid(i,j)=1 otherwise 
% solid(i,j)=0. 
%  
NI=100; 
x=zeros(NI+1,2); % allocate correct sized array 
y=zeros(NI+1,2); % allocate correct sized array 
xcen=zeros(NI,1); % allocate correct sized array 
solid=zeros(NI,1); % allocate correct sized array 
dx=2; % cell length in x direction 
dy=1; % cell length in y direction (arbitrary) 
for i=1:NI+1 
    for j=1:2 
        x(i,j)=(i-1)*dx; 
        y(i,j)=(j-1)*dy; 
    end % of j loop 
end % of i loop 
% find cell centres by averaging coordinates of vertices  
% (this works for a general structured mesh) 
for i=1:NI 
     xcen(i,1)=(x(i,1)+x(i+1,1)+x(i+1,2)+x(i,2))/4; 
end % of i loop 
end % freadmesh 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
function A=fcellarea(x,y)   
% Verification:  verified. 
% In a uniform Cartesian mesh cell area = dx dy  
dx=x(2,1)-x(1,1); 
dy=y(1,2)-y(1,1);    
A=dx*dy 
end % fcellarea 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
function S=fsidevectors(x,y) 
% Verification: not verified 
% For each cell: 
% right side vector = dy i + 0 j  
% left side vector = -dy i + 0 j 
% 
S=zeros(2,2); 
dx=x(2,1)-x(1,1); 
dy=y(1,2)-y(1,1); 
% Right side vectors 
S(1,1)=dy; 
S(1,2)=0; 
% Left side vectors 
S(2,1)=-dy; 
S(2,2)=0; 
end % fsidevectors 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
function u=finitialu(xcen,t,vx)  
% Verification:  verified 
% Inserts initial or exact u values at time t.  
[NI,NJ]=size(xcen); 
u=zeros(NI,1); 
% Initial 1D Gaussian function based on the centre, x=xc, of the  
% computational domain. 
xmax=max(max(xcen));  % max x value in mesh 
xmin=min(min(xcen));  % min x value in mesh 
xc=(xmax+xmin)/2;  % approx x coord of mesh centre 
% 
cutoff=(xmax-xc)/2;  % cut off radius for Gaussian 
for i=1:NI 
        x=xcen(i,1)-vx*t; 
        d=abs((x-xc)); % distance from centre 
        if (d<cutoff) 
           u(i,1)=exp(-0.01*d^2);  % Gaussian 
        else 
           u(i,1)=0; 
        end 
end % of i loop 
end % finitialu 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function IH=finterfaceflux(v,u0,i) 
% Verification:  not verified 
% Calculates right and left interface fluxes for each cell. 
% Flux H depends on U, i.e. H = H(U).  
% For the 1D linear advection equation, H(U) = vx U i + 0 j. 
%  
%  
%% The following QUAD scheme  
%  obtains interface u values at i+1/2 by quadratic interpolation using 
%  the 2 neighbouring upwind cell centre values and the neighbouring 
%  downwind cell centre value.  For vx > 0 these are indexed by i-1, i and 
%  i+1. 
% it can be shown that the interpolated interface value of u at i+1/2 is 
% (-u(i-1)+6*u(i)+3*u(i+1))/8; 
% 
%  so  IHright = v ((-u0(i-1,1)+6*u0(i,1)+3*u0(i+1,1))/8; 
%  and IHleft  = v ((-u0(i-2,1)+6*u0(i-1,1)+3*u0(i,1))/8; 
%  so 2 left hand and 1 right hand ghost values for u are needed 
% 
IH=zeros(2,2); % array to store left and right interface flux vectors 
% 
vx=v(1); 
vy=v(2); 
% Right 
IH(1,1)=vx*((-u0(i-1,1)+6*u0(i,1)+3*u0(i+1,1))/8);  % i component of right interface flux 
IH(1,2)=0;           % j component of right interface flux 
% Left 
IH(2,1)=vx*((-u0(i-2,1)+6*u0(i-1,1)+3*u0(i,1))/8);  % i component of left interface flux 
IH(2,2)=0;             % j component of left interface flux 
end % fIflux 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
function outarray=finsertghostcells(inarray) 
% Verification:  not verified 
% For the quad scheme 2 ghost cells are needed at left edge of domain,  
% and 1 at right.  array is embedded into (m+3)x1 array of zeros.   
% Hence computational indices go from i=3 to i=m+2. 
[m,n]=size(inarray); 
dummy=zeros(m+3,1); 
dummy(3:m+2,1)=inarray; 
outarray=dummy; 
end % finsertghostcells 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function u=fbcs(u) 
% Verification:  not verified 
% Implements boundary conditions using ghost cells indexed by i=1,2,m+2. 
[m,n]=size(u); 
NI=m-3; 
% Dirichlet: u=0 at each end. 
% Don't need to do anything as u values in ghost cells are already zero. 
end % bcs 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function dt=fcalcdt(A,S,v) 
% Verification:  not verified 
% Finds allowable time step dt using heuristic formula. 
F=0.01; % tuning factor (F=1 gives exact results) 
rightside=[S(1,1),S(1,2)]; 
dt=A/abs(dot(v,rightside)); % heuristic formula 
dt=F*dt; 
end  % fcalcdt 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function fdrawmesh(x,y,solid) 
% Verification:  not verified 
% Description:  Draws a structured 2D finite volume mesh and fills in any 
% solid cells.  
% Date structures: 
% The mesh has NI cells in the i direction and NJ cells in the j direction. 
% The x and y coordinates of the lower left hand corner of cell (i,j) are 
% held in arrays x and y respectively which are both NI+1 by NJ+1.  In 
% this way the 4 vertices of cell (i,j) are (x(i),y(j)), (x(i+1),y(j)), 
% (x(i+1),y(j+1)) and (x(i),y(j+1)).  solid is an NI by NJ array which  
% flags solid cells.  If cell (i,j) is solid then solid(i,j)=1 otherwise 
% solid(i,j)=0. 
%  
[m,n]=size(x); 
NI=m-1; % number of cells in i direction 
NJ=1; % number of cells in j direction 
% 
% Plot the mesh 
hold on   % keeps all plots on the same axes 
% draw lines in the i direction 
for i=1:NI+1 
    plot(x(i,:),y(i,:)) 
end 
% draw lines in the j direction 
for j=1:NJ+1 
    plot(x(:,j),y(:,j))    
end 
title('computational mesh') 
xlabel('x') 
ylabel('y') 
% Fill in solid cells 
for i=1:NI 
    for j=1:NJ 
        if (solid(i,j)==1) 
            solidx=[x(i,j),x(i+1,j),x(i+1,j+1),x(i,j+1),x(i,j)]; 
            solidy=[y(i,j),y(i+1,j),y(i+1,j+1),y(i,j+1),y(i,j)]; 
            fill(solidx,solidy,'k')       
        end 
    end % of j loop 
end % of i loop 
end % fdrawmesh 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
function fplotresults(xcen,u0,uinitial,uexact) 
% Verification:  not verified 
% Dispays results on regular Cartesian mesh 
[NI,NJ]=size(xcen); 
u0=u0(3:NI+2,1); % extract computational values 
plot(xcen,uinitial,'k--',xcen,uexact,xcen,u0,'k.') 
%plot(xcen,u0) 
title('plots of initial profile (--), Quad numerical(.) and exact solutions (solid line)') 
xlabel('x [m]') 
ylabel('U [kg/m]') 
end % fplotresults 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 

